It is well accepted from non-perturbative Monte Carlo lattice simulations that \(SU(N)\)  Yang–Mills gauge theories in the absence of fundamental matter fields undergo a deconfining phase transition at a certain critical temperature [1]}, [2]}. This transition corresponds to the breaking of a global \(\mathbb {Z}_N\)  center symmetry when the Euclidean temporal direction is compactified on a circle, with circumference proportional to the inverse temperature, see e.g. [3]}, [4]}. The vacuum expectation value of the Polyakov loop [5]} serves as an order parameter for this symmetry, and has as such inspired an ongoing research activity into its dynamics, see e.g. [6]}, [7]}, [8]}, [9]}.
Even in the presence of dynamical quark degrees of freedom, in which case the center symmetry is broken explicitly, the Polyakov loop remains the best observable to capture the cross-over transition, see [10]}, [11]} for ruling lattice QCD estimates. Since the transition temperature is of the order of the scale at which the considered gauge theories, including QCD, become strongly coupled, it is a highly challenging endeavour to get reliable estimates for the Polyakov loop correlators, including its vacuum expectation value, analytically. This is further complicated by the non-local nature of the loop. These features highlight the sheer importance of lattice gauge theories to allow for a fully non-perturbative computational framework. Nonetheless, analytical takes are still desirable to offer a complementary view at the same physics, in particular as lattice simulations do also face difficulties when the physically relevant small quark mass limit must be taken, next to the issue of potentially catastrophic sign oscillations at finite density [12]}, [13]}.
